Nonetheless, if the axion–vector coupling is large enough, it might become feasible to generate a sufficiently large hypermagnetic helicity, alongside a corresponding set of fermionic charge asymmetries, in models of postinflationary axion evolution.
An important aspect in this case consists in the fact that an oscillating axion field is also subject to Hubble friction, which leads to damped oscillations around the potential minimum.
It is therefore possible to generate a nonvanishing net helicity from asymmetric oscillations; see, e.g., Refs. [1]}, [2]}, which discuss axion-driven magnetogenesis at the time of inflaton oscillations during reheating.
Meanwhile, it is challenging to realize efficient axion-driven magnetogenesis at later times in the cosmological evolution, e.g., in scenarios where the axion field begins to oscillate during the radiation-dominated era, without requiring a prohibitively large axion–vector coupling.
To overcome this problem, one could imagine that the onset of axion oscillations is delayed by an additional time-dependent contribution to the axion mass, or one could consider axion oscillations during an early stage of matter domination.
A third option would be to replace the axion field by a complex field rolling in the complex plane [3]}, [4]}, [5]}, [6]}, similarly as in the Affleck–Dine mechanism [7]}, [8]}, [9]}, such that the sign of the axion velocity remains unchanged during magnetogenesis and helicity production becomes more efficient.
